I need help with this problem: Use the five properties of exponents to simplify each of the expressions.
(y^5)^3(y^3)^2/(y^4)^4
(y^5)^3(y^3)^2/(y^4)^4
Power of a power: When an exponent has an exponent, both are multiplied and applied to the base of the first.
Example: (y^5)^3 = (y^15)
When multiplying terms with exponents, the exponents are added:
(y^15)(y^6) = (y^21)
When dividing, they're subtracted:
(y^21)/(y^16) = y^5
I think this is correct; Bob or the like will come along and beat me if it isn't. :P
law of indices
To simplify the given expression, we can make use of the following properties of exponents:
1. Product of Powers Property: When raising a power to another power, you multiply the exponents.
2. Quotient of Powers Property: When dividing two powers with the same base, you subtract the exponents.
3. Power of a Power Property: To raise a power to another power, you multiply the exponents.
4. Power of a Product Property: To raise a product to a power, you raise each factor to that power.
5. Power of One Property: Any number raised to the power of 1 is equal to the number itself.
Let's simplify the given expression step by step:
1. Start with the expression: (y^5)^3(y^3)^2/(y^4)^4
2. Apply the Power of a Power property for (y^5)^3: y^(5*3) = y^15
3. Apply the Power of a Power property for (y^3)^2: y^(3*2) = y^6
4. Apply the Power of a Power property for (y^4)^4: y^(4*4) = y^16
5. Substitute the simplified expressions back into the original expression: y^15 * y^6 / y^16
6. Apply the Quotient of Powers property for y^15 / y^16: y^(15 - 16) = y^(-1)
7. Rewrite y^(-1) as 1/y: 1/y
Therefore, the simplified form of the expression (y^5)^3(y^3)^2/(y^4)^4 is 1/y.